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16ИСТОРИЧЕСКИЕ НАУКИ
APPLICATION OF ELEMENTS OF HIGHER MATHEMATICS IN PARACHUTE TECHNOLOGY Pulatova M.I.
Pulatova Manzura Iskhakovna - Candidate of Physical and Mathematical Sciences,
Associate Professor, DEPARTMENT HIGHER MATHEMATICS, BUKHARA ENGINEERING TECHNOLOGICAL INSTITUTE, BUKHARA, REPUBLIC OF UZBEKISTAN
Abstract: the article discusses some examples of the use of elements of higher mathematics in solving theoretical issues of aerodynamics and dynamics of parachute systems; boundary-value flow problems associated with the integral equation.
For the numerical solution of these equations, the method of discrete singularities was used. It is shown how modern parachute technology is diverse and for its production the efforts of various specialists, including mathematicians, are required.
Keywords: historical information, creation of a parachute, aeronautics, parachute technique, parachute aerodynamics, potential flow around the canopy, method of discrete features, design.
UDC 532.25 / 09
A parachute is a mechanical device designed to reduce the speed at which a person or a load falls. The braking effect is mainly due to the shape and dimensions of the dome .
The first information about the parachute into Europe came from China , according to which another 1 306 in honor of the coronation and m n eratora brave owl pw al and jumping from the high towers using paper umbrellas. The first unit project, with which you can safely descend from any height, found in the "astrological th code e " L eonardo da Vinci, where the chapter "On the flying and move the AI in the air" are drawing and calculation of the ball square, cw total of four triangular panels u, stretched along the lower perimeter in a wooden square frame. Calculate the optimal area of the tent made've face them Italian, remind so modern. Creating a project parachute Da Vinci refers to the 1 486- 151 0 GG.
Only a century later, in 1616 city of Yugoslav scientist Fau hundred Vrancic made the parachute with a square dome-like about CPC da Vinci, but was not able to test it.
History svidetels tvuet also about trying to escape from the places Zak for prison by a fixture th, made of sheets and spanned mentioned on a rigid frame.
End X Y III Art. considered the beginning of the era of aeronautics , when in 1784, for the first time in the skies of France, Gerard Michel Montgolfier and Sebastian Lenormand raised a balloon . However, even in 1777 city of Etienne Montgolfier with parachute proper design made b lagopoluchny jump from the roof of his home. At the end e of the X Y III Art. next to the square and domes , round structures also appear on a rigid frame, which makes them bulky and inconvenient to use. These parachutes were attached to the balloon shell in an expanded form. During this period, parachute jumps were performed mainly for the entertainment of numerous spectators.
In 1797, the aeronaut André Jacques Garnerin first used a parachute as a rescue device. During the flight over Paris, he left the balloon at an altitude of 1000m. which immediately exploded, and landed safely in front of the astonished audience. The jump of Garneren was observed by the physicist and astronomer Lalande , who, after the necessary calculations, advised the balloonist to make a small hole in the center of the canopy to eliminate the swinging of the parachute canopy during the descent.
With this improvement, which was later called the pole hole, Garnerin made the jump on October 24, 1800 and landed on the Champ de Mars. His brother Jean Baptiste Garnerin created a parachute design without a rigid frame, thus changing the weight and dimensions of the device. Batista's adopted daughter Elizabeth Garnere n . Began jumping with a parachute of this design in 1791.
At the same time, there are a number of attempts to fundamentally change the design of the parachute. These include the "flying cloak" of Defontan, the "inverted" version of the dome of K. Keili , etc. However, these projects were not successful enough.
Since 1783, flights began on balloons filled with hot air or hydrogen, which allowed the balloonists to significantly increase the flight altitude. In this regard, the need for a parachute as a rescue vehicle has increased, which entailed the need to improve parachute systems.
In the 80s of the XIX century. German inventor Hermann Lateman has created a new scheme for opening the parachute in the process of putting it into operation. For the first time, the dome began to fit into a special bag in the form of a sleeve, which is then attached to the balloon gondola in the form of a rolled roll. Lateman made a number of successful jumps with his parachute. Unfortunately, the designer died during one of the jumps. His business was continued by his colleague and wife Ketchen Paulus, who for several years made balloon flights and parachute jumps.
In Russia, interest in aeronautics and parachute jumping has arisen for a long time. In 1 856 in Well Urnaliev "Sea Collection" in the well-known scientist of K.I. Konstantinova setting out the history of the parachute, design features ns and ra account settings parachute systems. In the early 90s, A.H. Regman, a scientist and home teacher of N. V. Zhukovsky , was engaged in the study of the stability of parachutes filled with air . Similar problems devoted his labor NF Yagn.
On November 9, 1903, the Wright brothers from the state of Northern California were the first in history to ascend into the sky in an apparatus heavier than air. Young aviation began to develop in many countries. Along with it, the number of accidents grew. The need to secure rescue funds ie became apparent. Those Paracha th you that when menilis on balloons were unfit us for use on airplanes. The slight modification made so that these parachutes could be used by aviators if necessary did not inspire confidence in them. Needs was received fundamentally different device to save the Well Life Span person if the airplane suffered a disaster.
A real revolution in parachute technique was made by the Russian inventor, artist of the Imperial Theater G.V. Kotelnikov. Neo bhodimo noted that at one time to the otelnikov b lestyasche graduated from artillery yskoe school in Kiev in the rank of officer and servi sludge F itomire, Poltava. Already during his service, he showed the talent of an inventor.
In 191 0 Kotelniki witnessed tragic th death of Russian pilot Leo M atsie vicha during the show -negative flight " F Armand" in St. Petersburg. He set out to create a reliable apparatus that is so necessary for aviators. Having familiarized himself with all the previous inventions, Kotelnikov realized that a device was needed that would be on the aviator all the time and would not interfere with it during the summer. In 1910-1912. he IMAGE l p Antsev parachute PK-I. Dome n arashyuta with straps fit into a special backpack, equipped with a revealing ustro th stvom. It is so wound u using L pits, forming boiling harness located on the back of years chica. Suspended th system would l and with a onstruirovan and thus the power of dynamic stroke, occurs when filling the dome with air, evenly distributed throughout the body.
Kotelnikov had to endure a long and stubborn struggle between the bureaucratic machine and the tsarist regime, seeking permission from high-ranking officials to test the invention. There were many skeptics who assert that at the moment of opening the parachute, the dynamic blow will be so strong that the person's arms, legs and head will come off. They could not be convinced by the calculations presented by the author and claiming the opposite. It was allowed as inventors to use mannequins with sand. Despite everything,
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Kotelnikov did not manage to convince everyone that a man should jump with his parachute. For the first time this jump was made in 1913 by Sergei Ossovsky in Paris. It was only during the First World War that Gleb Evgenievich was instructed to lead the specialists for the production of parachutes as a rescue tool for aviators who flew in combat conditions.
Only after the revolution in the 1920s did Kotelnikov create versions of his parachute -RK-2 and RK-3. The basic principles of the design and commissioning of the Kotelnikov parachute have been preserved in our time. One of the central problems of the aerodynamics of a parachute is the determination of the force action of the medium on the canopy, which leads to the calculation of the drag of the plane streamlined and spherical.
The problems of the aerodynamics and dynamics of the parachute are multiparametric; to solve it, it is necessary to develop mathematical models, taking into account the resistance, filling, and stability of the parachute canopy.
Based on the theory of the potential flow properties equations H ave- Stokc and the experimental data, calculated distribution of pressure on the inner side of the dome.
This pressure
PB=P - Q q, (1)
where is the pressure in the liquid far from the surface of the dome, the velocity head, and the dimensionless number of discharge depending on the shape of the dome. The determination of the gas resistance for a given one is related to the problem of the potential flow around an open surface, which leads to a singular integral equation solved by the discrete vortex method. The problem of potential flow around the dome gives the dependence of the drag coefficient C on the number of vacuum and the opening angle of the dome
For canopies with a parachute configuration (more ), the flow rate under the canopy is very low and the pressure drop is almost constant across the canopy
P * (1 + Q ) q, (2)
From where
C*1 + Q (3)
The value differs markedly from the constant only near the edge, where the pressure drop rapidly decreases to zero.
It is shown , that for a given depending dome shape design scheme of the time allows us to find the pressure distribution over the surface. In this regard, the Neumann problem for the Laplace equation is solved , which leads to a singular integral equation of the first kind.
For small domes, the shape is not known in advance and depends on the distribution of the pressure drop , which, in turn, depends on the shape of the shell. Solutions developed method, according to which the surface of the dome up Rob iruetsya family of functions depending on the parameter type. To determine them, a system of nonlinear homogeneous differential equations is obtained, which implies the dependence of the shell shape on time.
It is assumed that the dome is an impenetrable shell of revolution that moves along the
axis. If is the velocity potential, away from the dome, that is, everything satisfies the Laplace equation
A p = 0 (4)
on the surface of the dome - the condition of no leakage
_ £ = (5) ,
where Ve - velocity of surface points, n - singular vector of surface's normal, — -derivative along the normal, V - is the speed of some point on the axis of symmetry, where
_ Ve=V + Va
Va - deformation rate.
With structural holes having axial symmetry (pole holes, slots), additional conditions are required with respect to the circulation of velocities. The simplest model of the dome is a weightless shell of revolution, in which there is no stress in the azimuthal direction, and in
the meridian direction it is inextensible. In this case, the equation for determination is obtained by the Galerkin method.
Spatial movement, questions of oscillations, stability of parachute systems were considered in their works by A.G. Byushgens, B. Ya. Lotinsky, B.A. Privalov. The parachute system was replaced by a system of rigid bodies with known aerodynamic characteristics; the shape of the filled canopy was allowed to remain unchanged during movement.
The use of such a model made it possible to find the drag coefficients of a filled dome in a certain range of angles of attack, and to analyze the oscillatory stability of a variable system. When solving these problems, two systems were used: a fixed inertial system in Cartesian coordinates and a mobile system with coordinates
For the dome of an open shell, the flow stall from the edge can be directed under the dome or behind it. A change in the stall direction leads to a sharp change in the aerodynamic characteristics of the system. It was shown in Ref. N that in a potential flow at the edge of the flow around the surface, the tangent to the meridial component of the fluid velocity takes infinite values. The direction of stall coincides with the direction of the transition of the potential flow velocity from to. If under the dome, and on the outer side of the dome near the edge, the flow stall occurs from under the dome, behind it a near wake zone with an almost constant value is formed, the theory of potential flow is inapplicable here. If under the dome, above the outer surface , then the flow stall is directed under the dome. Thus, it has been proved that the flow crisis arises when the edge of the dome satisfies the Chaplygin-Zhukovsky condition
wi = 0 (6)
After the onset of the crisis and the restructuring of the current field, deterioration of the parachute's braking characteristics may occur.
The boundary value problems of the flow around the surface of revolution of the potential theory method are related to an integral equation. In studies [6], a one-parameter approximation of the dome shape is used, which is determined by one generalized coordinate. The boundary value problems of the flow are reduced to two problems:
- solution of a singular integral equation;
- solution of an integro-differential singular equation with a kernel of the Cauchy type.
Numbers for flax Foot solving these equations was used IU Todd discrete features.
Experiments and observations indicate that there is a pulsating instability of the parachute. The issues of the influence of the main parameters of the parachute system (line length, pole hole size, Newton's number, etc.) on stability have not been sufficiently taught.
Modern parachute technology is diverse both in its purpose and in its structure. Parachutes are used not only to save the lives of pilots, but also for the purpose of landing military units, as braking devices when landing high-speed aircraft and space ships, for delivering people and cargo to inaccessible terrain, for performing sports jumps, etc. Depending on the purpose, these parachutes differ in design and material part.
In accordance with the purpose, the following types are distinguished:
- brake parachute;
- for landing cargo;
- to solve auxiliary tasks;
- to drop people.
Braking parachute has a long history. It was developed at the beginning of the 20th century. by a Russian designer, and was originally intended for braking cars. In this form, the idea did not take root, but in the late 1930s. it begins to take root in aviation.
Today, the braking parachute is part of the braking system of fighters that have a high landing speed and a short landing distance, for example, on warships. When approaching the runway of such aircraft, one braking parachute with one or more canopies is ejected from the tail of the fuselage. Its use can shorten the braking distance by 30%. In addition, the braking parachute is used when landing space challengers.
Special parachute systems, consisting of one or several canopies, are used to land cargo ejected from aircraft. If necessary, such systems can be equipped with reluctance motors that give an additional braking impulse before direct contact with the ground. Such parachute systems are also used for the descent of spacecraft to the ground. Parachutes for auxiliary tasks include those that are integral parts of parachute systems:
- exhaust , which pulls the main or spare canopy;
- stabilizing, which, in addition to pulling, have the function of stabilizing the dropped object;
- supporters who ensure the correct process of opening the other parachute.
Most of the parachute systems exist for the landing of people.
The following types of parachutes are used to safely land people:
- training;
- rescue;
- special . destination;
- landing;
- gliding shell parachute systems (sports).
The main types are gliding shell parachute systems ("wing") and landing (round) parachutes.
Army parachutes are of 2 types: round and square.
The canopy of a round landing parachute is a polygon, which, when filled with air, takes the shape of a hemisphere. The dome has a cutout (or less dense fabric) in the center. Round landing parachute systems (for example, D-5, D-6, D-10) have the following altitude characteristics:
- maximum drop height - 8 km.
- normal working height - 800-1200 m.
- the minimum drop height is 200 m with stabilization for 3 s and descent on a filled dome for at least 10 s.
Round landing parachutes are poorly controlled. They have approximately the same vertical and horizontal speed (5 m / s). Weight:
- 13.8 kg (D-5);
- 11.5 kg (D-6);
- 11.7 (D-10).
Square parachutes (for example, the Russian "Leaf" D-12, the American T-11) have additional slots in the canopy, which gives them better maneuverability and allows the parachutist to control horizontal movement. The rate of descent is up to 4 m / s. Horizontal speed - up to 5 m/s.
Training parachutes are used as intermediate parachutes for the transition from landing to sport parachutes . They, like the landing, have round domes, but are equipped with additional slots and valves that allow the skydiver to influence the horizontal movement and train landing accuracy.
The most popular training option is D-1-5U. It is he who is used when making the first independent jumps in parachute clubs. By pulling one of the steering lines, this model makes a full 360 ° C turn in 18 seconds. He is well managed.
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Average sink rates ( m/s):
- horizontal - 2.47;
- vertical - 5.11.
The minimum ejection height from D-1-5U is 150 m with immediate deployment. Maximum ejection height - 2200 m. Other training models: P1-U; T-4; UT-15. Having characteristics similar to those of the D-1-5U, these models are even more maneuverable: they make a full turn in 5 s, 6.5 s and 12 s, respectively. In addition, they are about 5 kg lighter than the D-1-5U.
Their production requires a lot of efforts of various kinds of specialists, including mathematicians.
References
1. Volmir A.S. "Shells in the ceiling of a liquid and the problem of aeroelasticity". M.: Nauka, 2007, 416 p.
2. Lobanov M.A., Churikov F.S. "Basic theories and calculations of parachutes". M.: Nauka, 1998.
3. Lisov I. Free flight. M.: "Young Guard" 1979, 221 p.
4. Rakhmatulin Kh.A. The theory of parachute deployment. Air Fleet Technique magazine, 1999.
5. Temnenko V.A. Influence of inertial parameters on the stability of parachute systems. Kiev, " Visha school" 2002.
6. Shevlyakov Yu.A., Tishchenko V.N., Temnenko V.A. "The dynamics of parachute systems". Kiev: "Vishcha school", 2002. 158 s.
